Brushless permanent magnet machine with axial modules of rotor magnetization skew and method of producing the same

ABSTRACT

An electrical machine having a machine output rating. The electrical machine including a shaft rotatable about an electrical machine axis, a rotor coupled to the shaft and rotating with the shaft, and a stator including a stator core. The rotor is configurable to include a first rotor portion having a relation to a first output rating and a second rotor portion having a relation to a second output rating. The stator core is configurable to be disposed adjacent to the first rotor portion relative to the machine axis when the machine output rating corresponds to the first output rating and adjacent to the second rotor portion when the machine output rating corresponds to the second output rating.

FIELD OF THE INVENTION

The invention relates to a brushless permanent magnet machine with axialmodules of rotor magnetization skew and method of producing the same.

BACKGROUND

Electrical machines, such as brushless permanent magnet (BLPM) motors,typically encounter problems with cogging and ripple torque, both ofwhich cause noise and vibration and can negatively affect the motorstarting performance. The sum of the cogging and ripple torquecomponents is defined as the electrical machine pulsating torque. Rippletorque is characterized as a cyclical variation in a delivered torque toa load caused by the interaction of the rotor magnetic field withharmonics in the stator current magnetomotive forces (mmf's). Coggingtorque describes the non-uniform torque as a result of the interactionof the rotor magnetization and angular variations in an air gappermeance (or reluctance) associated with the shape of the slots of thestator. By definition, no stator excitation is involved in coggingtorque production. There is a demand for an electric motor thatminimizes the effect of cogging and ripple torque and that exhibitssmooth operation. Further, there is a demand for an electrical machinehaving a rotor operable to provide a plurality of power or motor ratingsfor a given motor frame, and thereby reduce tooling costs and inventory.Further, there is a demand for an electrical machine to be easilyconfigurable for operation with different combinations of the number ofphases and poles.

SUMMARY

In one embodiment, the invention provides an electrical machine having amachine output rating. The electrical machine includes a shaft rotatableabout an axis, a rotor mounted or coupled to the shaft and rotating withthe shaft, and a stator including a stator core and windings. The rotoris configurable to include a first rotor portion having a relation to afirst output rating and a second rotor portion having a relation to asecond output rating. The stator core is configurable to be disposedadjacent to the first rotor portion when the machine output ratingcorresponds to the first output rating and adjacent to the second rotorportion when the machine output rating corresponds to the second outputrating.

In another embodiment, the invention provides an electrical machine thatcan be set up for operation in one of a plurality of modes including afirst mode wherein the electrical machine includes a first machineoutput rating and a second mode where the electrical machine includes asecond machine output rating. The second machine output rating isdifferent than the first machine output rating. The electrical machineincludes a shaft rotatable about an electrical machine axis, a rotormounted or coupled to the shaft and rotating with the shaft, and astator including a stator core and windings. The rotor is a first rotorin the first mode and a second rotor in the second mode. The first rotorhas a first rotor length and/or a first magnetization patterncorresponding to the first machine output rating. The second rotor has asecond rotor length and/or a second magnetization pattern correspondingto the second output rating. The stator core is a first core in thefirst mode and a second core in the second mode. The first core has afirst core length corresponding to the first machine output rating, andthe second core has a second core length corresponding to the secondmachine output rating.

In yet another embodiment, the invention provides an electrical machinehaving a shaft rotatable about an electrical machine axis, a rotormounted or coupled to the shaft and rotating with the shaft, and astator including a plurality of stator teeth. Each stator tooth includesone or more channels along a surface adjacent to the rotor. The channelincludes one of a trapezoidal shape, and a curvilinear shape.

In another embodiment, the invention provides a method of manufacturingan electrical machine having a stator and a rotor. The method includesthe acts of determining a desired output rating from a plurality ofoutput ratings; determining a length of the stator, the length having arelation to the desired output rating; determining a length of therotor, the length having a relation to the desired output rating;producing the stator; providing a magnetizer configured to magnetize therotor into a plurality of sections; and producing the rotor. The act ofproducing the rotor includes magnetizing the rotor to include a firstsection when the desired output rating corresponds to the first outputrating and magnetizing the rotor to include the first section and asecond section when the desired output rating corresponds to the secondoutput rating.

Other aspects of the invention will become apparent by consideration ofthe detailed description and accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is partial exploded view of the stator and rotor of a brushlesspermanent magnet electrical machine.

FIG. 2 is a longitudinal view of one construction of the rotor of FIG.1.

FIG. 3 is a longitudinal view of another construction of the rotor ofFIG. 1.

FIG. 4 is a longitudinal view of yet another construction of the rotorof FIG. 1.

FIG. 5 is a longitudinal view of another construction of the rotor ofFIG. 1.

FIG. 6 is a cross-sectional view of a stator core and a rotor capable ofbeing used in the electrical machine of FIG. 1.

FIG. 7 is a partial cross-sectional view of a portion of a stator corecapable of being used in the electrical machine of FIG. 1.

FIG. 8 is a partial cross-sectional view of a portion of a stator corecapable of being used in the electrical machine of FIG. 1.

FIG. 9 is a partial cross-sectional view of a portion of a stator corecapable of being used in the electrical machine of FIG. 1.

FIG. 10 a is a combination longitudinal view of a rotor andlongitudinal-sectional view of a stator, which has the same core lengthas the rotor.

FIG. 10 b is a combination longitudinal view of a rotor andlongitudinal-sectional sectional view of a stator, which has a shortercore length than the rotor and spacers are used to axially align thestator and the rotor.

FIG. 11 is an example of a stator-winding pattern in a double-layerarrangement with compact coils for an 18-slot, 12-pole, 3-phase machine.

FIG. 12 is an example of a stator-winding pattern in a single-layerarrangement with compact coils for an 18-slot, 12-pole, 3-phase machine.

FIG. 13 is an isometric view showing the geometry used to define an arcof magnetization skew (p) on the rotor.

DETAILED DESCRIPTION

Before any embodiments of the invention are explained in detail, it isto be understood that the invention is not limited in its application tothe details of construction and the arrangement of components set forthin the following description or illustrated in the following drawings.The invention is capable of other embodiments and of being practiced orof being carried out in various ways. Also, it is to be understood thatthe phraseology and terminology used herein is for the purpose ofdescription and should not be regarded as limiting. The use of“including,” “comprising,” or “having” and variations thereof herein ismeant to encompass the items listed thereafter and equivalents thereofas well as additional items. The terms “connected,” “coupled,” and“mounted” and variations thereof herein are used broadly and, unlessotherwise stated, encompass both direct and indirect connections,couplings, and mountings. In addition, the terms connected and coupledand variations thereof herein are not restricted to physical andmechanical connections or couplings.

FIG. 1 is a partial exploded view of the stator and rotor of oneconstruction of an electrical machine (e.g., motor, generator, etc.).For FIG. 1, the electrical machine is a motor 10 having a rotor 15 and astator 20. The rotor 15 is coupled to a shaft 17. In general, the stator20 receives electrical power, and produces a magnetic field in responsethereto. The magnetic field of the stator 20 interacts with a magneticfield of the rotor 15 to produce mechanical power on the shaft 17. Theinvention below refers to the electrical motor 10, however the inventionis not limited to the motor 10.

The rotor 15 includes a plurality of magnetic poles 25 of alternatingpolarity exhibited on a surface of a rotor core 30. The rotor core 30includes laminations (e.g., magnetic steel laminations), and/or solidmaterial (e.g., a solid magnetic steel core), and/or compressed powderedmaterial (e.g., compressed powder of magnetic steel). One constructionof the rotor 15 includes a sheet of permanent magnet (e.g., hardmagnetic) material disposed on the rotor core 30. Another constructionof the rotor 15 can include a plurality of strips of permanent magnetmaterial attached (e.g., with adhesive) around the core 30. Thepermanent magnet material can be magnetized by a magnetizer to provide aplurality of alternating magnetic poles. Additionally, the number ofmagnetic strips can be different than the number of rotor magneticpoles. Yet another construction of the rotor 15 contains blocks ofpermanent magnet material placed inside the rotor core 30.

The description of the invention is not limited to a particularmechanical construction, geometry, or position of the rotor 15. Forexample, FIG. 1 shows the rotor 15 located inside and separated by aradial air gap from the stator 20. In another construction, the rotor 15can be positioned radially exterior to the stator 20 (i.e., the machineis an external- or outer-rotor machine.)

One method to reduce cogging and ripple torque is skewing themagnetization of the magnetic poles 25 with respect to the stator 20.Alternatively, stator teeth of the stator 20 can be skewed with respectto the rotor magnetization. The optimal arc of skew in the magnetizationof the rotor is dependent on the electrical machine topology andparticular machine design. As shown in FIGS. 1-5, the “magnetization” ofthe rotor 15 refers to the line pattern 31 along the length of the rotor15 delineating alternating magnetic poles 25 on the rotor core 30. Eventhough a rotor 15 of the invention can include any number of alternatingmagnetic poles 25, FIGS. 2-5 show only one line pattern along the rotor15 for the sake of simplicity.

FIG. 13 illustrates the geometrical concepts involved in defining themagnetization skew of the rotor. The arc of magnetization skew can bedefined as the arc (i), measured in radians in between the longitudinallines 32 and 33 (see FIG. 2) on the rotor surface facing the air-gap,which separates the stator and the rotor.

FIG. 2 is a schematic diagram of one construction of the rotor 15divided into a plurality of axial sections 55 (e.g., 70, 71, and 72)along the rotational axis 50 of the rotor 15. The number of axialsections 55 can vary and is not limiting on the invention. An axialsection 55 refers to a portion of the rotor 15 differentiated byimaginary lines 60. Imaginary lines 60 prefer to locations on the rotor15 where the direction of skew of the magnetization pattern 31 changes.One construction of the rotor 15 includes alternating magnetic poleswith substantially the same arc of magnetization skew (β) along eachaxial section 55, resulting in a herringbone pattern of magnetization.The length of each axial section 55 can vary. The arc of magnetizationskew is generally the same for each axial section 55 in order to ensurethe continuity of the magnetic poles, and is selected such as tominimize cogging and ripple torque. However, the outer axial sections(denoted by 95 and 100 in FIG. 5) can have a different arc ofmagnetization skew as it will be explained later.

The design (e.g., the length and magnetization pattern) of the rotorrelates to the desired output rating (e.g., power rating in horsepoweror torque and speed rating) of the electrical machine, where the desiredpower output rating is one of multiple ratings for the profile of thestator and rotor. Herein, the cross-sectional profile of the stator androtor refers to the cross-sectional geometry of the stator core 105 androtor 15. For example, FIG. 6 shows the profile of one construction ofthe stator core 105 and rotor 15 of the invention. The electricalmachine 10, in one construction, provides a plurality of output ratingsusing the same profile of the stator core 105 by varying the length(e.g., the number of laminations in the stack) of the stator core inmagnetic interaction with a respective combination of axial sections 55of the rotor 15. This aspect of the invention reduces tooling costs andinventory. In some constructions, when varying by design the statorstack length, the winding pattern is kept the same and the number ofturns and wire size are changed in order to match the electrical powersupply conditions, the desired output rating, and other designrequirements, such as the copper fill factor.

One method of providing the herringbone magnetization pattern (seeFIG. 1) on the rotor 15 includes the use of a magnetizer and amagnetizing fixture. Normally, a particular magnetizing fixture isrequired for an electrical machine having a particular length of rotor15. The rotor 15 of the invention allows the same magnetizer andmagnetizing fixture to be used for multiple output power ratings,thereby reducing tooling costs.

FIG. 2 shows one construction of the rotor 15 including three axialsections 70, 71, and 72. The stator 20 interacts with one or more of thethree axial sections 70, 71, and 72 to provide multiple output ratingsfor the profile of the motor. The first axial section 70 includesmagnetic poles aligned with a first skew direction, the second axialsection 71 includes magnetic poles aligned with a second skew direction,and the third axial section 72 includes magnetic poles aligned with thefirst skew direction. The second axial section 71 interacts with a firststator 20 to provide a minimum rating for the profile of the motor(e.g., a one-half horsepower output). A combination of the first 70 andsecond 71 axial sections interact with a second stator 20 to provide anintermediate rating for the profile of the motor (e.g., a three-quarterhorsepower output). A combination of all three axial sections 70, 71,and 72 interact with a third stator 20 to provide a maximum rating forthe profile of the motor (e.g., a one horsepower output). Thisconstruction of the rotor 15 includes fewer changes in skew directionalong the rotor with respect to the description of the otherconstructions given below, such that the magnetizing fixture includes amore simple and accurate magnetization pattern. Making the first 70 andthird 72 axial sections of the same length and arc of magnetization skew(β), contributes to the axial magnetic symmetry of the motor having themaximum rating for the profile of the motor. However, with aconstruction of the rotor 15 as shown in FIG. 2, characterized asincluding a number of axial sections equal to the number of possibleoutput ratings within the profile of the motor 10, the axial symmetry ofa motor of an intermediate output rating is not necessarily guaranteed.FIG. 3 is a schematic diagram of yet another construction of the rotor15. The construction shown in FIG. 3 is characterized as including anumber of axial sections equal to double the number of possible outputratings within the profile of the motor 10. For example and as shown inFIG. 3, the rotor 15 includes six axial sections 75, 76, 77, 78, 79, and80 operable to provide three power output ratings. The third 77 andfourth 78 axial sections interact with a first stator 20 to provide aminimum rating for the profile of the motor (e.g., one-half horsepoweroutput). A combination of the second 76, third 77, fourth 78, and fifth79 axial sections interact with a second stator 20 to provide anintermediate rating for the profile of the motor (e.g., a three-quarterhorsepower output). All six axial sections 75, 76, 77, 78, 79, and 80interact with a third stator 20 to provide a maximum rating for theprofile of the motor (e.g., a one horsepower output). The first, second,and third stators 20 described above include the same profile of statorcore 105, but can differ in the length (e.g., number of laminations) ofthe stator core to provide the desired output rating.

FIG. 4 is a schematic diagram of yet another construction of the rotor15 of the invention. This construction is characterized by the number ofaxial sections equal to double minus one the number of possible outputratings within the profile of the motor 10. For example, FIG. 4 showsthe rotor 15 including five axial sections 85, 86, 87, 88, and 89operable to provide three power output ratings. The third 87 axialsection interacts with a first stator 20 to provide a minimum rating forthe profile of the motor (e.g., one-half horsepower output). Acombination of the second 86, third 87, and fourth 88 axial sectionsinteract with a second stator 20 to provide an intermediate rating forthe profile of the motor (e.g., a three-quarter horsepower poweroutput). A combination of the all five axial sections 85, 86, 87, 88,and 89 interact with a third stator 20 to provide a provide a maximumrating for the profile of the motor (e.g., a one horsepower output).Each of the sections 85, 86, 87, 88, and 89 can be of different length.The first, second, and third stators 20 described above include the sameprofile of stator core 105, but can differ in the length (e.g., numberof laminations) of the stator core to provide the desired output rating.

The constructions of the invention shown in FIG. 3 and FIG. 4 providemore freedom to vary by design the lengths of the axial sections andimprove the axial symmetry of the motor. In the rotor construction shownin FIG. 3, the axial symmetry of the motor is improved if the first 75and sixth 80, the second 76 and the fifth 79, the third 77 and thefourth 78 axial sections, have, respectively, equal length and arc ofmagnetization skew. In the rotor construction shown in FIG. 4 the axialsymmetry of the motor is improved if the first 85 and the fifth 89, thesecond 86 and the fourth 88 axial sections have, respectively, equallength and arc of magnetization skew. In the rotor construction shown inFIG. 5 the axial symmetry of the motor is improved if the first 95 andthe sixth 100, the second 96 and the fifth 99 axial sections, and thethird 97 and the fourth 98 axial sections have, respectively, equallength and arc of magnetization skew.

One construction of a rotor 15 design includes a first one or more axialsections in relation to a first output rating (P₁) (e.g., one-halfhorsepower output). The first one or more axial sections 55 have a firstlength (L₁). A ratio of the first length L₁ of the first one or moreaxial sections 55 divided by a maximum length (L_(m)) of the rotor, usedfor a maximum rating (P_(m)) for the profile of the motor, is in a rangeof ¾ to 1½ times the ratio of the power ratings (P₁/P_(m)), with apreferred range of ¾ to 1¼ times the ratio of the power ratings(P₁/P_(m)). This range of power and length ratio provides the designerwith freedom to design for a desired output rating by trading off, onone hand the motor size and cost, and on the other hand motorefficiency. The rotor 15 can also include a second one or more axialsections in relation to an intermediate rating for the profile of themotor (P₁) (e.g., a half horsepower output). The second one or moreaxial sections have a second length (L₁), and the second length includesthe first length. A ratio of the second length of the second one or moreaxial sections divided by a maximum length of the rotor (L_(m)) is in arange of ¾ to 1½ of the ratio of the power ratings (P₁/P_(m)), with apreferred range of ¾ to 1¼ times the ratio of the power ratings(P₁/P_(m)).

The total number of axial sections and the total number of ratings for agiven motor profile are not limiting on the invention. Therefore,generally speaking, a rotor 15 design includes one or more axialsections in relation to a first output rating (P_(x)) (e.g., one-halfhorsepower output, ¾ horsepower output, etc.). The one or more axialsections 55 have a first total length (L_(x)). A ratio of the firsttotal length L_(x) of the one or more axial sections 55 divided by amaximum length (L_(m)) of the rotor, used for a maximum rating (P_(m))for the cross-sectional profile, is in a range of ¾ to 1½ times theratio of the power ratings (P_(x)/P_(m)), with a preferred range of ¾ to1¼ times the ratio of the power ratings (P_(x)/P_(m)).

In addition to reducing cogging and ripple torque, the arc ofmagnetization skew also affects the specific torque output (e.g., torqueper unit axial length at a given current) of the motor 10. In general,the torque output or power rating decreases as the arc of magnetizationskew increases. Reducing the arc of magnetization skew can increase themotor torque output per axial length. Accordingly, at the penalty ofincreasing the cogging and ripple torque, reducing the arc ofmagnetization skew allows shortening of the axial length of the rotor 15and maintaining a desired power output of the motor 10. Shortening theaxial length of rotor 15 reduces material costs.

FIG. 5 shows yet another construction of the rotor 15 including innersections 96, 97, 98, and 99 having substantially equal arc ofmagnetization skew, and outer sections 95 and 100 having a lesser arc ofmagnetization skew with respect to the magnetization skew of the innersections 96, 97, 98, 99. The inner sections 96, 97, 98, 99 include thesame arc of magnetization skew to enhance continuity and symmetry. Theouter sections 95 and 100 have a lesser arc of magnetization skew (β) toenhance the output rating of the electrical machine with use of all sixaxial sections 95, 96, 97, 98, 99, and 100.

For FIGS. 2-5 described above, the value and number of output ratingscan vary and is not limiting on the invention. In addition, theincremental difference between output ratings related to one or acombination of axial sections 55 can vary and is not limiting on theinvention.

In one construction of the invention, the electrical machine includesthe rotor 15 having a plurality of axial sections as shown in one ofFIGS. 2-5. The rotor 15 provides multiple output ratings for the profileof the machine. This construction of the electrical machine uses thesame rotor 15 in electrical machines of varying output ratings, therebyreducing the part count and the inventory required for producing a rangeof motors of different output ratings using the same motor profile.

In another construction of the electrical machine, one or more axialsections 55 of the rotor 15 are not present when the desired outputrating of the electrical machine is less than the maximum power outputrating for the machine to be produced using the same stator coreprofile. For example, for the electrical machine having a desired outputrating of one-half horsepower, the rotor axial sections 70 and 72 ofFIG. 2 are not necessary. This construction of the electrical machineallows the use of the same magnetizer to magnetize the rotor 15 having arange of output ratings within the profile of the machine. In addition,this construction reduces the material waste (e.g., 70 and 72 of FIG. 2)of the rotor 15.

Various designs of stator 20 can be used to interact with eachconstruction of the rotor 15 described above and shown in FIGS. 2-5. Thefollowing is a description of one construction of the invention thatincludes the rotor 15 disposed radially from the stator 20. Withreference to FIG. 1, the stator 20 includes a stator core 105 having aplurality of stator teeth 110 and stator windings 112. In oneconstruction, the stator core 105 includes a stack of magnetic steellaminations or sheets. In other constructions, the stator core 105 isformed from a solid block of magnetic material, such as compacted powderof magnetic steel. The stator windings 112 include electrical conductorsplaced in the slots 120 (FIG. 6) and around the plurality of teeth 110.Other constructions and types of the stator core 105 and stator windings112 known to those skilled in the art can be used and are not limitingon the invention.

Electrical current flowing through the stator windings 112 produces amagnetic field that interacts with the magnetization of the rotor 15 toprovide torque to the rotor 15 and shaft 17. The electrical current canbe an (m) phase alternating current (AC), where (m) is an integergreater than or equal to two. The electrical current can have varioustypes of waveforms (e.g., square wave, quasi-sine wave, etc). The statorwindings 112 receive electrical current from an electrical drive circuit(not shown). One construction of the electrical drive circuit includes acontroller and an inverter with one or more power electronic switches(e.g., MOSFET, IGBT) to vary the flow of electrical current to thewindings dependent on various electrical machine operating parameters(e.g., speed, load, rotor position, etc.). To determine the position ofthe rotor 15, the control circuit includes, in some constructions, asensor (e.g., Hall effect device, encoder, etc.) to provide the controlcircuit with a signal representative of the rotor position.Alternatively, the control circuit can estimate the rotor positionthrough what is commonly referred as a sensorless control. Theelectrical drive circuit can include other components and circuitconstructions known to those skilled in the art and is not limiting onthe invention.

FIG. 6 shows a cross-sectional profile of a motor cross-sectionperpendicular to axis 50 used in one motor construction (the statorwindings 112 are not shown in FIG. 6). The stator core 105 includes theplurality of stator teeth 110, slots 120, and a back iron portion 115.Each of the plurality of stator slots 120 receives one or more statorcoils, the assembly of which constitutes the stator windings 112. Thestator windings receive a multi-phase electrical current, where thenumber of phases (m) is an integer greater than or equal to two. Thenumber (t) of stator teeth 110 equals the number of slots 120, where (t)is an integer. A slot 120 is defined by the space between adjacentstator teeth 110. The rotor 15 is produced, in one construction, byfixing three arc shaped magnets 26 on a rotor core 30. Other rotordesigns and constructions are also possible as mentioned previously. Amagnetizer is used to produce on the rotor 15 a number (p) ofalternating magnetic poles that interact with the stator 20, where (p)is an even (i.e., divisible by 2) integer greater than or equal to two.The stator core 105 includes a ratio of the number of stator teeth tomagnetic poles (t/p) equal to (m/2) or (m/4).

The stator core 105 having the above-described construction (see FIG. 6)can be used to design and manufacture motors with various (m) electricphases, with windings 112 composed of compact coils (see the windingpatterns in FIG. 11 and FIG. 12) and rotors having poles (p). Forexample, a stator core 105 having a same cross sectional profile with anumber (t) of stator teeth 110 can be used, in principle, to producemotors with (m) phases or an increased number of phases (km). In orderto maintain the same (t/p) ratio, the number of poles can be reduced to(p/k), and therefore (k) can be any integer for which (p/k) is an eveninteger greater than or equal to two. Alternatively, the number ofphases can be decreased from (m) to an integer (m/k), where (k) is anyinteger for which (m/k) is an integer greater than or equal to two. Inorder to maintain the same (t/p) ratio, the number of magnetic poles canbe increased to (kp).

To provide a stator 20 with (m) symmetrical electric phases, within eachphase the compact coils, belonging to the phase, are connected such thatconsecutive phases are placed at a mechanical angle of (4π/(mp))radians. For any number of phases (m), the number (t) of teeth andnumber (p) of poles is designed so that their ratio (t/p) is equal to(m/2) or (m/4). The number of teeth per pole and phase (t/p/m) istherefore a design constant, equal to ½ or ¼ respectively, and thereforefor constant air-gap magnetic loading (i.e. flux density), the magneticflux per tooth remains constant. Therefore, the stator teeth 110 can beoptimally designed for any number of phases.

In some constructions of the machine, it is generally desired for theback iron portion 115 (see FIG. 6) to operate at approximately the samemagnetic loading as the teeth 110. To equalize the magnetic loading, aminimum width of the back iron portion 115 can equal half the value ofthe product of the number of teeth per pole times the tooth width. Theminimum width (w_(y)) of the back iron is defined as the minimumdistance between the top of a slot 120 and a circle with the center onthe rotational axis 50 and a radius equal to the minimum distancebetween the rotational axis 50 and any of the flat surfaces from theoutside surface of the stator core 105 (see FIG. 6). The number of teethper pole (t/p) can be an integer or a fractional number. Limits on theminimum width of the back iron portion 115 include manufacturability andincreased mmf drop and core losses related to back-iron flux density.For a design with an increased number of phases (m), the number of poles(p) is decreased by design in order to maintain a constant ratio (t/p)for a given lamination. Lowering by design the number of poles (p)results in an increased magnetic pole pitch and, for the same air-gapmagnetic loading, an increased magnetic flux density in the back ironportion 115. Lowering by design the number of poles (p), also results ina decrease of the fundamental frequency of the magnetic field for agiven rotational speed of electrical machine and limits core losses inthe back-iron portion 115. Finite element analysis that considers theabove-described parameters indicates the width of the back iron portion115 granges between (1½-4½) times the product of the number of teeth perpole (t/p) divided by 2 and times the tooth width (w_(t)).

One construction of the stator windings 112 includes a double layerarrangement of compact coils (FIG. 11), which are placed around eachtooth (i.e. the coils have a pitch of 1-slot). In this double layerarrangement, each slot is shared by two coil sides, each of the coilsides belonging to a different coil and phase. The two coil sidessharing a slot can be placed side by side or one on top of the other.The double-layer winding pattern for an example 18-slot, 12-pole,3-phase winding is shown in FIG. 11. Following the rules set above, fora given stator core and a winding with compact coils and a double layerpattern, the coil connections, the number of turns per coil and the wiresize can be modified by design in order for the machine to operate withany number of phases (m) and poles (p) for which (t/p) is equal to (m/2)or (m/4).

Another construction of the windings 112 includes a single layerarrangement of compact coils (FIG. 12), which are placed around everyother tooth (i.e. the coils have a pitch of 1-slot and are only placedaround half the number of teeth). In this single layer arrangement, eachslot contains only one coil side. The single layer winding pattern foran example 18-slot, 12-pole, 3-phase winding is shown in FIG. 12.Following the rules set above, for a given stator core and a windingwith compact coils and a single layer pattern, the coil connections, thenumber of turns per coil and the wire size can be modified by design inorder for the machine to operate with any number of phases (m) and poles(p) for which (t/p) is equal to (m/2) or (m/4). In comparison with adouble layer winding with compact coils, a single layer winding withcompact coils has only half the number of coils but the per phaseend-winding is generally longer.

The phase windings of the stator 20 are symmetrically and equidistantlydistributed at an angle of (2π/m) electrical radians or (4π/(mp))mechanical radians. A symmetrical (m) phase system of currents flowingthrough the stator windings produces a magnetomotive force (mmf) with aspace electrical fundamental harmonic of the mechanical order (p/2). Themmf also includes space harmonics of the electrical order (2 km−1) and(2 km+1), where k is an integer larger or equal to one. When theelectrical machine couples to a load, the mmf harmonics cause rippletorque, an undesired machine characteristic described above. Theamplitude of the mmf harmonic increases as its harmonic order decreases.The amplitude of the lower-order mmf harmonics (2 m−1) and (2 m+1) canbe significant and their reduction ensures a smooth motor operation.

With simple compact windings, built according to the previousdescription, conventional means of reducing the mmf harmonics, such asshort-pitching the winding are not available. Instead, an optimalmagnetization skew is determined and implemented to reduce the mmfharmonics and the ripple torque, as well as the cogging torque.

The skew factor for a v-th electrical order mmf space harmonic is givenby the equation: (k_(sv)=4 sin(vpβ/4)/(vpβ)), where the arc ofmagnetization skew (β) is measured in radians on the rotor surfacefacing the air-gap (see FIG. 2). A harmonic is completely eliminated ifthe argument of the sine function satisfies the equation (vpβ/4=nπ),where (n) is an integer larger or equal to zero. For an mmf harmonic ofthe electrical order (v=2 km−1) the previous equation is equivalent to(β=4 nπ/(p(2 km−1))) and for an mmf harmonic of the electrical order(v=2 km+1) the previous equation is equivalent to (β=4 nπ/(p(2 km+1))).For increasing values of (n) and/or (k), both arrays (4 nπ/(p(2 km−1)))and (4 nπ/(p(2 km+1))) converge to (2π/(pm)).

Therefore, to reduce both (2 km−1) and (2 km+1) orders of mmf spaceharmonics, one construction of the motor 10 includes the stator 20having a ratio of stator teeth 110 per magnetic pole (t/p) equal to(m/2) and the rotor 15 including an arc of magnetization skew(β=2π/(pm)) measured in radians on the rotor surface facing the air-gap.Another construction of the motor 10 includes the stator 20 having aratio of stator teeth 110 per pole (t/p) equal to (m/4) and the rotor 15including an arc of magnetization skew (β=2π/(pm)) measured in radianson the rotor surface facing the air-gap.

A typical manufacturing technique to provide a double layer statorwinding with compact coils includes use of a needle or gun winder. Asubstantially large opening of the stator slot 120 is beneficial towardsthe air-gap in order to allow the needle of the winder to be insertedinto the slot.

A typical manufacturing technique to provide a single layer statorwinding with compact coils includes use of an insertion winder. Asubstantially large opening of the stator slot 120 is required in orderto allow the conductors to be inserted into the slot. Other types andtechniques known to those in the art to provide the stator windings 112of the stator 20 can be used.

A relatively large opening of slot 120 increases the ease of insertionof the needle winder and of the conductors of the windings,respectively. An opening of the slot 120 suitably large to becost-effective for automatic winding manufacturing includes a rangegreater than ⅙th of a tooth pitch. Tooth pitch is the distance betweenadjacent centerlines 135 (see FIGS. 7-9) of teeth 110. The slots 120create a variation of the permeance of the air-gap between the rotor 15and the stator 20. The variation in air-gap permeance interacts with themagnetic field of the rotor 15 to cause cogging torque. As noted above,cogging torque is an undesired characteristic of electrical machines andits minimization, by reducing the variation of the air-gap permeance,while still maintaining a slot opening suitably large for volumemanufacturing technologies.

FIGS. 7, 8, and 9 show a construction of the stator core 105 thatincludes “dummy” channels 130 in the stator teeth 110. The dummychannels 130 reduce the amplitude and, for certain motor designs, canincrease the frequency of the cogging torque, as it will be shown in thefollowing. The shape and dimensions of each dummy channel 130 are variedby design to provide a more symmetrical variation of the cogging torqueversus rotor position. Constructions of the stator 20 include a suitablecore 105 having one or two dummy channels 130 per tooth 110. Of course,the stator 20 of the invention can include more dummy channels 130 andis not limiting on the invention.

The number of equivalent openings of slots 120 of the stator 20 includesthe number of slots 120 and the number of dummy channels 130 g(see FIGS.7-9). By adding a number (d) of dummy channels 130 at the free ends ofeach tooth 110, where (d) is an integer greater than or equal to zero,the number of equivalent slot openings towards the air-gap increasesfrom the number (t) to ([d+1]t). The frequency of the cogging torque isequal to the least common multiple of the number ([d+1]t) of equivalentslot openings and the number of poles (p).

For (t/p=m/2), mathematical induction proves the following: Phases PolesCogging frequency (m) (p) Dummy channels (d) (Hz) 2k + 1 4j 0 or 1 mp2k + 1 4j 2 3mpwhere (k) and (j) are integers greater than or equal to one. In each ofthe above cases, the arc of magnetization skew (β) equal to (2π/(mp)),as measured in radians in between the longitudinal lines 32 and 33 onthe rotor surface facing the air-gap (see FIG. 13), causes a reductionof both the cogging torque and torque ripple.

For (t/p=m/4), mathematical induction proves the following: Phases PolesCogging frequency (m) (p) dummy channels (d) (Hz) 2k 2j 1 mp 2k + 1 2j 0or 1 mp 2k + 1 2j 2 3mpwhere (k) and (j) are integers greater than or equal to one. In each ofthe above cases, the arc of magnetization skew (β) is equal to(2π/(mp)), as measured in radians in between the longitudinal lines 32and 33 on the rotor surface facing the air-gap (see FIG. 13), causes areduction of both the cogging torque and torque ripple.

FIG. 7 shows one construction of a stator tooth 110 including a dummychannel 130 having a centerline located at the middle of the tooth 110and coinciding with the tooth centerline 135. In the construction fromFIG. 7, the dummy channel 130 is generally trapezoidal-shaped and ischaracterized by a top width of channel (w_(n)) a bottom width ofchannel (W_(b)), and the side angle (α). The width of the slot opening(w_(o)) is designed to a minimum value for which cost-effectivemanufacturing of the stator winding is achieved and the cogging torqueis low. The height of the slot opening (h_(o)), and the dimensions ofthe dummy channels (w_(n)), (W_(b)) and (α) are designed to optimize themachine from a magnetic and mechanical point of view.

Finite element analysis of the electromagnetic field indicates aconstruction of the channel 130 of FIG. 7, including a top width (w_(n))ranging from (0.5 w_(o))≦(w_(n))≦(1.5 w_(o)), a bottom width (w_(b))ranging from (0.3 w_(o))≦(w_(b))≦(1.2 w_(o)) and the side angle (α)ranging from (30°)≦(α)≦(135°), controls the local level of magneticsaturation in the tooth tip, modifies the air-gap magnetic permeance,reduces the cogging torque, and improves the symmetry of the coggingtorque variation against rotor position. Therefore, the cogging torqueis substantially reduced in a motor which has, in addition, the rotormagnetization skewed with the optimal arc of skew (β) previouslydetermined.

FIG. 8 shows a second construction of a stator tooth 110 including twodummy channels 130. The dummy channels are located so that theircenterlines 137 are dividing the slot pitch, which is contained inbetween the centerlines 136 of two adjacent slots, in three intervals ofapproximately equal length. In the construction from FIG. 8, the dummychannel 130 is generally trapezoidal-shaped and is characterized by atop width of channel (w_(n)), a bottom width of channel (w_(b)), a sideangle (α), and yet another side angle (γ). The width of the slot opening(w_(o)) is designed to a minimum value for which cost-effectivemanufacturing of the stator winding is achieved and the cogging torqueis low. The height of the slot opening (h_(o)), and the dimensions ofthe dummy channels (w_(n)), (w_(b)), (α), and (γ) are designed tooptimize the machine from a magnetic and mechanical point of view.

Finite element analysis of the electromagnetic field indicates aconstruction of the channel 130 of FIG. 8, including a top width (w_(n))ranging from (0.5 w_(o))≦(w_(n))≦(1.5 w_(o)), a bottom width (w_(b))ranging from (0.3 w_(o))≦(w_(b))≦(w_(o)) and the side angles (α) and (γ)ranging from (30°)≦(α)≦(90°) and (30°)≦(γ)≦(90°), controls the locallevel of magnetic saturation in the tooth tip, modifies the air-gapmagnetic permeance, reduces the cogging torque, and improves thesymmetry of the cogging torque variation against rotor position.Therefore, the cogging torque is substantially reduced in a motor whichhas, in addition, the rotor magnetization skewed in the optimal arc ofskew (β) previously determined. For a construction of a stator 30including two dummy channels 130 per teeth 110, space limitations canlimit the value of the side angles (α) and (γ) to be equal or belowninety degrees.

FIG. 9 shows another construction of a stator tooth having twocurvilinear shaped dummy channels 130. The dummy channels are located sothat their centerlines 137 are dividing the slot pitch, which iscontained in between the centerlines 136 of two adjacent slots, in threeintervals of approximately equal length. The opening of the dummychannels 130 towards the air-gap is equal to the opening (w_(o)) of theslot 120. The curvilinear shape follows that of an arc of the circlewith the center on the respective dummy channel centerline and adiameter larger or equal to ¾ of (w_(o)) and smaller or equal to 1½ of(w_(o)). This shape and dimensions of the dummy channels reduce thecogging torque and increase the durability of the punching die used formanufacturing stator laminations.

Having described constructions of the electrical machine, a method ofassembling one construction of the electrical machine will now bedescribed. It is envisioned that the method may be modified for otherconstructions. Furthermore, it is envisioned that not all of the actsbelow may be required, that some of the acts may be modified, or thatthe order of the acts may vary.

A designer provides the rotor 15 having the plurality of alternatingmagnetic poles. The rotor 15 is divided into a plurality of portionsalong the longitudinal axis 50. The plurality of portions can include afirst portion related to a first output rating (e.g., one-halfhorsepower), a second portion relating to a second output rating (e.g.,three-quarter horsepower), and a third portion relating to a thirdoutput rating (e.g., one horsepower).

Each of the portions is divided into one or more axial sections 55(e.g., axial sections 70, 71, and 72 in FIG. 2). Each axial section 55includes a respective arc of magnetization skew (β) of the alternatingmagnetic poles in relation to the first, second, and third outputratings of the electrical machine. The arc of magnetization skew (β) ismeasured on the rotor surface facing the air-gap, as shown in FIG. 2. Amagnetizer is used to provide the magnetization of the axial sections ofthe rotor. This method of constructing the rotor 15 allows a commonmagnetizer to be used to provide the magnetization of the rotor for aplurality of desired output ratings of the electrical machine, therebyreducing tooling costs. In one construction of the electrical machine,the end axial sections 55 that are not needed to provide the desiredoutput rating are not included with the rotor 15 and therefore also thematerial cost is reduced. In another construction, all axial sections 55(e.g., sections in relation and not in relation to the desired outputrating) of the rotor 15 can be retained in the assembly of the rotor 15.This second construction is advantageous if, for example, the cost ofthe rotor material from the end axial sections that are not necessarilyrequired in relation to the desired output rating is smaller than thecost savings achieved by maintaining an inventory with only a reducednumber of rotor dimensions.

Using a uniform profile of the stator 20, the designer determines thelength of the stator core 105 to interact with the rotor 15 to providethe desired output rating. For example, with a laminated construction ofthe stator core 105, the designer selects a stack length of laminationsof magnetic material to provide the desired output rating. The statorcore 105 is wound with windings 112 designed for the electrical supplyconditions, the stator core length, the rotor length, and the desiredmotor output. The manufacturing operator aligns the stator 20 with therotor 15, so that the axial centerline of the stator core 105 coincideswith the axial centerline of the rotor 15 and no side-pull axial forcesare exhibited due to stator-rotor misalignment (see FIG. 10 a).Referring to FIG. 10 b, if the rotor 15 includes other end axialsections not in relation to the desired output rating (e.g. 75 and 80),additional end axial spacers 150 can be added to help align the statorcore 105 with the rotor 15. For example, referring to FIGS. 3 and 10 b,if the desired output rating is three-quarter horsepower, the stator 20would be aligned with the axial sections 76, 77, 78, and 79. Two spacers150 are used to cover the axial sections 75 and 80, respectively, of therotor 15 not in relation to the desired three-quarter horsepowermachine. The end-spacers can enhance support of assembly of theelectrical machine in a uniform housing. The axial length of spacers 150can vary with the constructions of the rotor 15 and stator 20 describedabove.

Thus, the invention provides, among other things, an electrical machinewith reduced cogging and torque ripple. Various features and advantagesof the invention are set forth in the following claims.

1-15. (Cancelled).
 16. An electrical machine having an output rating(P_(x)), the electrical machine comprising: a shaft rotatable about anaxis; a stator including a stator core and windings, the stator corehaving a fixed cross-sectional profile with respect to the axis; and arotor coupled to the shaft and adapted to magnetically interact with thestator, the rotor having a periphery, the rotor being configured toinclude three or more axial sections, each of the axial sections havinga respective magnetization pattern of alternating magnetic poles skewedwith respect to the axis, the magnetization pattern for each of theaxial sections being different than any adjacent axial section, thethree or more axial sections defining a total length (L_(x)) in theaxial direction with respect to the axis, where (L_(x)) satisfies therelationship (0.75(P_(x)/P_(m))≦(L_(x)/L_(m))≦1.5(P_(x)/P_(m))), where(P_(m)) is a chosen maximum output rating for an electrical machinebuilt with a stator core having the fixed cross-sectional profile, where(L_(m)) is the length corresponding to the length of the rotor for themaximum output rating (P_(m)), where (P_(x)) is less than the maximumoutput rating (P_(m)), and where each of the axial sections define anarc of skew (β) at the periphery, (β) being the same for each of theaxial sections. 17-19. (Cancelled).
 20. The electrical machine of claim16, wherein the axial sections substantially define a herringbonepattern in the axial direction.
 21. (Cancelled).
 22. The electricalmachine of claim 16, wherein the lengths of each axial section areapproximately the same. 23-45. (Cancelled).
 46. An electrical machinecomprising: a shaft rotatable about an axis; a stator including a statorcore and windings; a rotor coupled to the shaft and adapted to interactwith the stator, the rotor including a center cross-section and one ormore permanent magnets configured to form at least four axial sections,each of the four axial sections having a magnetization pattern ofalternating magnetic poles that are skewed with respect to the axisalong substantially straight lines and having an arc of magnetizationskew that is substantially the same as the other axial sections, the atleast four axial sections including a first axial section having a firstmagnetization direction and a first axial length, a second axial sectiondisposed adjacent to the first axial section, the second axial sectionhaving a second magnetization direction, having a second axial lengththat is substantially the same as the first axial length, and beingsymmetric to the first axial section with respect to the centercross-section, a third axial section disposed adjacent to the firstaxial section, the third axial section having a third magnetizationdirection that is different than the first magnetization direction andhaving a third axial length, and a fourth axial section disposedadjacent to the second axial section, the fourth axial section having afourth magnetization direction that is different than the thirdmagnetization direction, having a fourth axial length that issubstantially the same as the third axial length, and being symmetric tothe third axial section with respect to the center cross-section; andwherein the first, second, third, and fourth magnetization directionsdefine a continuous zig-zag pattern in the axial direction.
 47. Theelectrical machine of claim 46, wherein a ratio range of the sum of thefirst axial length and the second axial length with the sum of the thirdaxial length and the fourth axial length is between 0.75 and 1.5. 48.The electrical machine of claim 46, wherein the first, second, third,and fourth magnetization directions substantially define a herringbonepattern in the axial direction.
 49. The electrical machine of claim 46,wherein the lengths of each axial section are approximately the same.50. The electrical machine of claim 46, wherein the sum of the firstaxial length and the second axial length is different than the sum ofthe third axial length and the fourth axial length.
 51. The electricalmachine of claim 46, wherein the at least four axial sections include afifth axial section disposed adjacent to the third axial section, thefifth axial section having a fifth magnetization direction that isdifferent than the third magnetization direction and having a fifthaxial length, and a sixth axial section disposed adjacent to the fourthaxial section, the sixth axial section having a sixth magnetizationdirection that is different than the fourth magnetization direction,having a sixth axial length that is substantially the same as the fifthaxial length, and being symmetric to the fifth axial section withrespect to the center cross-section; and wherein the first, second,third, fourth, fifth, and sixth magnetization directions define acontinuous zig-zag pattern in the axial direction.
 52. The electricalmachine of claim 51, wherein the fifth and sixth axial sections have anarc of magnetization skew that is substantially the same as the arc ofmagnetization skew of the first, second, third, and fourth axialsections.
 53. The electrical machine of claim 46, wherein the firstmagnetization direction is the same as the third magnetizationdirection.
 54. An electrical machine comprising: a shaft rotatable aboutan axis; a stator including a stator core and windings; a rotor coupledto the shaft and adapted to interact with the stator, the rotorincluding a center cross-section and one or more permanent magnetsconfigured to form at least three axial sections, each of the threeaxial sections having a magnetization pattern of alternating magneticpoles that are skewed with respect to the axis along substantiallystraight lines and having an arc of magnetization skew that issubstantially the same as the other axial sections, the at least threeaxial sections including a first axial section having a firstmagnetization direction and a first axial length, the centercross-section of the first axial section being the same as the centercross-section of the rotor, a second axial section disposed adjacent tothe first axial section, the second axial section having a secondmagnetization direction that is different than the first magnetizationdirection and having a second axial length, and a third axial sectiondisposed adjacent to the first axial section, the third axial sectionhaving a third magnetization direction that is different than the firstmagnetization direction, having a third axial length that issubstantially the same as the second axial length, and being symmetricto the second axial section with respect to the center cross-section;and wherein the first, second, and third magnetization directions definea continuous zig-zag pattern in the axial direction.
 55. The electricalmachine of claim 54, wherein a ratio range of the first axial lengthwith the sum of the second axial length and the third axial length isbetween 0.75 and 1.5.
 56. The electrical machine of claim 54, whereinthe first, second, and third magnetization directions substantiallydefine a herringbone pattern in the axial direction.
 57. The electricalmachine of claim 54, wherein the first axial length is approximately thesame as the sum of the second axial length and the third axial length.58. The electrical machine of claim 54, wherein first axial length isdifferent than the sum of the second axial length and the third axiallength.
 59. The electrical machine of claim 54, wherein the at leastfour axial sections include a fourth axial section disposed adjacent tothe second axial section, the fourth axial section having a fourthmagnetization direction that is different than the second magnetizationdirection and having a fourth axial length, and a fifth axial sectiondisposed adjacent to the third axial section, the fifth axial sectionhaving a fifth magnetization direction different that is than the fifthmagnetization direction, having a fifth axial length that issubstantially the same as the fourth axial length, and being symmetricto the fourth axial section with respect to the center cross-section;and wherein the first, second, third, fourth, and fifth magnetizationdirections define a continuous zig-zag pattern in the axial direction.60. The electrical machine of claim 59, wherein the fourth and fifthaxial sections have an arc of magnetization skew that is substantiallythe same as the arc of magnetization skew of the first, second, andthird axial sections.
 61. An apparatus having a family of electricalmachines, the apparatus comprising: a housing; a first electricalmachine disposed in the housing, the first electrical machine comprisinga first shaft rotatable about a first axis, a first stator including afirst stator core and first windings, the first stator core having across-sectional profile with respect to the first axis, a first rotorcoupled to the first shaft and adapted to magnetically interact with thefirst stator, the first rotor including a first center cross-section anda first one or more permanent magnets configured to form two axialsections, each of the two axial sections having a magnetization patternof alternating magnetic poles that are skewed with respect to the firstaxis, the two axial sections including a first axial section having afirst magnetization direction and a first axial length, a second axialsection disposed adjacent to the first axial section, the second axialsection having a second magnetization direction, having a second axiallength that is substantially the same as the first axial length, andbeing symmetric to the first axial section with respect to the firstcenter cross-section; a second electrical machine disposed in thehousing, the second electrical machine comprising a second shaftrotatable about a second axis, a second stator including a second statorcore and second windings, the second stator core having thecross-sectional profile with respect to the second axis, a second rotorcoupled to the second shaft and adapted to magnetically interact withthe second stator, the second rotor including a second centercross-section and a second one or more permanent magnets configured toform four axial sections, each of the four axial sections having arespective magnetization pattern of alternating magnetic poles that areskewed with respect to the second axis, the four axial sectionsincluding a third axial section having the first magnetization directionand the first axial length, a fourth axial section disposed adjacent tothe third axial section, the fourth axial section having the secondmagnetization direction, having the second axial length, and beingsymmetric to the third axial section with respect to the second centercross-section, a fifth axial section disposed adjacent to the thirdaxial section, the fifth axial section having a third magnetizationdirection that is different than the first magnetization direction andhaving a third axial length, and a sixth axial section disposed adjacentto the fourth axial section, the sixth axial section having a fourthmagnetization direction that is different than the second magnetizationdirection, having a fourth axial length that is substantially the sameas the third axial length, and being symmetric to the fifth axialsection with respect to the second center cross-section; wherein thefirst, second, third, and fourth magnetization directions define acontinuous zig-zag.
 62. The apparatus of claim 61, wherein a ratio rangeof the sum of the first axial length and the second axial length withthe sum of the third axial length and the fourth axial length is between0.75 and 1.5.
 63. The apparatus of claim 61, wherein the first, second,third, and fourth magnetization directions define a continuousherringbone pattern in the axial direction.
 64. The apparatus of claim61, wherein the lengths of each axial section are approximately thesame.
 65. The apparatus of claim 61, wherein the sum of the first axiallength and the second axial length is different than the sum of thethird axial length and the fourth axial length.
 66. The apparatus ofclaim 61, and further comprising: a third electrical machine disposed inthe housing, the third electrical machine comprising a third shaftrotatable about a third axis, a third stator including a third statorcore and third windings, the third stator core having thecross-sectional profile with respect to the third axis, a third rotorcoupled to the third shaft and adapted to magnetically interact with thethird stator, the third rotor including a third center cross-section anda third one or more permanent magnets configured to form six axialsections, each of the six axial sections having a respectivemagnetization pattern of alternating magnetic poles that are skewed withrespect to the third axis, the six axial sections including a seventhaxial section having the first magnetization direction and the firstaxial length, an eighth axial section disposed adjacent to the seventhaxial section, the eighth axial section having the second magnetizationdirection, having the second axial length, and being symmetric to theseventh axial section with respect to the third center cross-section, aninth axial section disposed adjacent to the seventh axial section, theninth axial section having the third magnetization direction and thethird axial length, and a tenth axial section disposed adjacent to theeighth axial section, the tenth axial section having the fourthmagnetization direction, having the fourth axial length, and beingsymmetric to the ninth axial section with respect to the second centercross-section; an eleventh axial section disposed adjacent to the ninthaxial section, the eleventh axial section having a fifth magnetizationdirection that is different than the third magnetization direction andhaving a fifth axial length, and a twelfth axial section disposedadjacent to the tenth axial section, the twelfth axial section having asixth magnetization direction that is different than the fourthmagnetization direction, having a sixth axial length that issubstantially the same as the fifth axial length, and being symmetric tothe eleventh axial section with respect to the third centercross-section; and wherein the first, second, third, fourth, fifth, andsixth magnetization directions define a continuous zig-zag.
 67. Theapparatus of claim 61, wherein the fifth and sixth axial sections havean arc of magnetization skew that is substantially the same as the arcof magnetization skew of the first, second, third, and fourth axialsections.
 68. An apparatus having a family of electrical machines, theapparatus comprising: a housing; a first electrical machine disposed inthe housing, the first electrical machine comprising a first shaftrotatable about a first axis, a first stator including a first statorcore and first windings, the first stator core having a cross-sectionalprofile with respect to the first axis, a first rotor coupled to thefirst shaft and adapted to magnetically interact with the first stator,the first rotor including a first center cross-section and a first oneor more permanent magnets configured to form one axial section, thefirst axial section having a magnetization pattern of alternatingmagnetic poles that are skewed with respect to the first axis and afirst axial length, the center cross-section of the first axial sectionbeing the same as the center cross-section of the first rotor; a secondelectrical machine disposed in the housing, the second electricalmachine comprising a second shaft rotatable about a second axis, asecond stator including a second stator core and second windings, thesecond stator core having the cross-sectional profile with respect tothe second axis, a second rotor coupled to the second shaft and adaptedto magnetically interact with the second stator, the second rotorincluding a second center cross-section and a second one or morepermanent magnets configured to form three axial sections, each of thethree axial sections having a respective magnetization pattern ofalternating magnetic poles that are skewed with respect to the secondaxis, the three axial sections including a second axial section havingthe first magnetization direction and the first axial length, the centercross-section of the second axial section being the same as the secondcenter cross-section of the second rotor, a third axial section disposedadjacent to the second axial section, the third axial section having asecond magnetization direction that is different than the firstmagnetization direction and having a second axial length, and a fourthaxial section disposed adjacent to the second axial section, the fourthaxial section having a third magnetization direction that is differentthan the first magnetization direction, having a third axial length thatis substantially the same as the second axial length, and beingsymmetric to the third axial section with respect to the second centercross-section; and wherein the first, second, and third magnetizationdirections define a continuous zig-zag.
 69. The apparatus of claim 68,wherein a ratio range of the first axial length with the sum of thesecond axial length and the third axial length is between 0.75 and 1.5.70. The apparatus of claim 68, wherein the first, second, and thirdmagnetization directions define a continuous herringbone pattern in theaxial direction.
 71. The apparatus of claim 68, wherein the first axiallength is approximately the same as the sum of the second axial lengthand the third axial length.
 72. The apparatus of claim 68, wherein firstaxial length is different than the sum of the second axial length andthe third axial length.
 73. The apparatus of claim 68, and furthercomprising: a third electrical machine disposed in the housing, thethird electrical machine comprising a third shaft rotatable about athird axis, a third stator including a third stator core and thirdwindings, the third stator core having the cross-sectional profile withrespect to the third axis, a third rotor coupled to the third shaft andadapted to magnetically interact with the third stator, the third rotorincluding a third center cross-section and a third one or more permanentmagnets configured to form five axial sections, each of the five axialsections having a magnetization pattern of alternating magnetic polesthat are skewed with respect to the third axis, the five axial sectionsincluding a seventh axial section having the first magnetizationdirection and the first axial length, an eighth axial section disposedadjacent to the seventh axial section, the eighth axial section havingthe second magnetization direction, having the second axial length, andbeing symmetric to the seventh axial section with respect to the thirdcenter cross-section, a ninth axial section disposed adjacent to theseventh axial section, the ninth axial section having the thirdmagnetization direction and the third axial length, and a tenth axialsection disposed adjacent to the eight axial section, the tenth axialsection having a fourth magnetization direction that is different thanthe second magnetization direction, and having a fourth length; aneleventh axial section disposed adjacent to the ninth axial section, theeleventh axial section having a fifth magnetization direction that isdifferent than the third magnetization direction, having a fifth axiallength that is substantially the same as the fourth axial length andbeing symmetric to the tenth axial section with respect to the thirdcenter cross-section; and wherein the first, second, third, fourth, andfifth magnetization directions define a continuous zig-zag.
 74. Theapparatus of claim 73, wherein the fourth and fifth axial sections havean arc of magnetization skew that is substantially the same as the arcof magnetization skew of the first, second, and third axial sections.